Question

In S.I. system, permeability has the unit
A. Weber meter-1 ampere-1
B. Weber meter ampere-1
C. Weber meter-1 ampere-2
D. Weber meter ampere

Answer 1

Hint: The permeability of free space can be derived by Biot-Savart’s law. It is an equation that describes the magnetic field generated by a constant current. The permeability depends upon the magnetic field, direction, length and the electric current.

Formula used:
Biot-Savart’s law is an equation that gives the magnetic field produced due to a current carrying segment. This law is given by:\[dB = \dfrac{{{\mu _0}Idl\sin \theta }}{{4\pi {r^2}}}\]

Complete step by step answer:
At present, the most widely used system of units is called the International System of units. It is abbreviated in SI.
The Biot-Savart’s law is given by:\[dB = \dfrac{{{\mu _0}Idl\sin \theta }}{{4\pi {r^2}}}\]
Rearranging this equation in terms of \[{\mu _0}\]
Thus, \[{\mu _0} = \dfrac{{dB\left( {4\pi {r^2}} \right)}}{{Idl\sin \theta }}\].
The unit for magnetic field is Tesla in the SI system of units. One Tesla is also equal to Weber per meter square \[\left( {\dfrac{{Wb}}{{{m^2}}}} \right)\]. Weber is actually the SI derived unit of magnetic flux. The unit for \[I\]is Ampere. Then, the units for \[{\mu _0}\]are
\[{\mu _0}\left( {units} \right) = \dfrac{{\dfrac{{Weber}}{{mete{r^2}}}\left( {mete{r^2}} \right)}}{{Ampere.meter}} = WeberAmper{e^{ - 1}}mete{r^{ - 1}}\]

So, the correct answer is “Option A”.

Note:
The unit Weber can also be written in SI base units as\[kg \cdot {m^2} \cdot {s^ - }^2 \cdot Amper{e^ - }^1\].
Therefore, the units of the permeability of free space can also be written as Newton per Ampere squared \[\left( {\dfrac{N}{{amper{e^2}}}} \right)\]. The universal constant value for the permeability of free space is \[4\pi \times {10^{ - 7}}WeberAmper{e^{ - 1}}mete{r^{ - 1}}\].